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arXiv:2606.00806

Computational Phase Transitions in Binary Compressed Sensing: Quantum Annealing Inside the Relaxation Gap

William Hahn, Natalia Romero

cs.ETquant-phSubmitted May 30, 2026• Announced Jun 2, 2026• 0 citations
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Abstract

We map the computational phase transition boundary in binary compressed sensing and identify a regime where D-Wave's quantum annealer recovers signals in a region where all tested classical methods fail, including Approximate Message Passing (AMP), which achieves the Bayes-optimal recovery threshold asymptotically for Gaussian matrices. In 19,775 experiments (n in {32, 64}, nine classical solvers, two D-Wave modes), we find that quantum annealing recovers sparse binary signals in the relaxation gap -- the regime below the Donoho-Tanner l1 phase transition where the l0 solution exists but convex relaxations fail. At n=32, k=5, m/n=0.19, D-Wave achieves 7% exact recovery while AMP and eight other solvers score 0% across 250 combined trials (Fisher exact p=0.018). At n=64, embedding overhead limits the QPU, but D-Wave's hybrid solver remains competitive with AMP. Energy landscape analysis reveals that the QUBO ground state contains the true signal, but incorrect solutions occupy shallower local basins that trap classical search -- a structure consistent with quantum tunneling dynamics. To our knowledge, this constitutes preliminary finite-size evidence that quantum annealing succeeds in a narrow regime where all tested classical methods, including the Bayes-optimal AMP, fail within a well-characterized combinatorial inference problem. Confirmation at larger n, higher trial counts, and with stronger classical controls remains an open problem.

Comments
7 pages, 4 figures

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We map the computational phase transition boundary in binary compressed sensing and identify a regime where D-Wave's quantum annealer recovers signals in a region where all tested classical methods fail, including Approximate Message Passing (AMP), which achieves the Bayes-optimal recovery threshold asymptotically for Gaussian matrices. In 19,775 experiments (n in {32, 64}, nine classical solvers, two D-Wave modes), we find that quantum annealing recovers sparse binary signals in the relaxation gap -- the regime below the Donoho-Tanner l1 phase transition where the l0 solution exists but convex relaxations fail. At n=32, k=5, m/n=0.19, D-Wave achieves 7% exact recovery while AMP and eight other solvers score 0% across 250 combined trials (Fisher exact p=0.018). At n=64, embedding overhead limits the QPU, but D-Wave's hybrid solver remains competitive with AMP. Energy landscape analysis reveals that the QUBO ground state contains the true signal, but incorrect solutions occupy shallower local basins that trap classical search -- a structure consistent with quantum tunneling dynamics. To our knowledge, this constitutes preliminary finite-size evidence that quantum annealing succeeds in a narrow regime where all tested classical methods, including the Bayes-optimal AMP, fail within a well-characterized combinatorial inference problem. Confirmation at larger n, higher trial counts, and with stronger classical controls remains an open problem.
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The focus of this work is to explore the use of quantum annealing solvers for the problem of phase unwrapping of synthetic aperture radar (SAR) images. Although solutions to this problem exist based on network programming, these techniques do not scale well to larger sized images. Our approach involves formulating the problem as a quadratic unconstrained binary optimization (QUBO) problem, which can be solved on a quantum annealer. Given that present embodiments of quantum annealers remain limited in the number of qubits they possess, we decompose the problem into a set of subproblems that can be solved individually. These individual solutions are close to optimal up to an integer constant, with one constant per subimage. In a second phase, these integer constants are determined as a solution to yet another QUBO problem. This basic idea is extended to several passes, where each pass results in an image which is subsequently decomposed to yet another set of subproblems until the resulting image can be accommodated by the annealer at hand. Additionally, we explore improvements to the method by decomposing the original image into overlapping subimages and ignoring the results on the overlapped (marginal) pixels. We test our approach with a variety of software-based QUBO solvers and on a variety of images, both synthetic and real. Additionally, we experiment using D-wave systems’ quantum annealer, the D-wave 2000Q_6 and developed an embedding method which, for our problem, yielded improved results. Our method resulted in high quality solutions, comparable to state-of-the-art phase-unwrapping solvers.
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We leverage the idea of a statistical ensemble to improve the quality of quantum annealing based binary compressive sensing. Since executing quantum machine instructions on a quantum annealer can result in an excited state, rather than the ground state of the given Hamiltonian, we use different penalty parameters to generate multiple distinct quadratic unconstrained binary optimization (QUBO) functions whose ground state(s) represent a potential solution of the original problem. We then employ the attained samples from minimizing all corresponding (different) QUBOs to estimate the solution of the problem of binary compressive sensing. Our experiments, on a D-Wave 2000Q quantum processor, demonstrated that the proposed ensemble scheme is notably less sensitive to the calibration of the penalty parameter that controls the trade-off between the feasibility and sparsity of recoveries.
Quantum annealing has emerged as a promising approach for solving NP-hard optimization problems, leveraging quantum phenomena such as quantum tunneling to navigate complex energy landscapes. However, the extent to which quantum tunneling contributes to performance enhancements compared to classical methods remains unclear. In this work, we present a comprehensive investigation combining experimental analysis, theoretical modeling, and algorithmic development to characterize when and why quantum annealing provides computational advantages. We introduce a novel methodology for generating synthetic Quadratic Unconstrained Binary Optimization (QUBO) problems with controlled gradient variance, enabling systematic investigation of landscape characteristics that favor quantum approaches. Our experimental evaluation encompasses four canonical NP-hard problems: Graph Partitioning, Max Cut, Number Partitioning, and Set Cover. Using D-Wave's Advantage2 quantum annealer with 4,400+ qubits, we compare quantum annealing against classical solvers including simulated annealing, stochastic gradient descent, and commercial optimization software. Our results demonstrate that quantum annealing shows measurable advantages when energy landscapes exhibit high gradient variance (>0.3> 0.3>0.3), suggesting that quantum tunneling effects are most beneficial for rugged optimization landscapes. We provide a theoretical justification for this empirical observation through a WKB-approximation-based model connecting gradient variance to barrier width and tunneling probability. This model achieves R2=0.90R^2=0.90R2=0.90 correlation with experimental data and yields quantitative predictions for quantum advantage thresholds.
The focus of this work is to explore the use of quantum annealing solvers for the problem of phase unwrapping of synthetic aperture radar (SAR) images. Although solutions to this problem exist based on network programming, these techniques do not scale well to larger-sized images. Our approach involves formulating the problem as a quadratic unconstrained binary optimization (QUBO) problem, which can be solved using a quantum annealer. Given that present embodiments of quantum annealers remain limited in the number of qubits they possess, we decompose the problem into a set of subproblems that can be solved individually. These individual solutions are close to optimal up to an integer constant, with one constant per sub-image. In a second phase, these integer constants are determined as a solution to yet another QUBO problem. We test our approach with a variety of software-based QUBO solvers and on a variety of images, both synthetic and real. Additionally, we experiment using D-Wave Systems's quantum annealer, the D-Wave 2000Q. The software-based solvers obtain high-quality solutions comparable to state-of-the-art phase-unwrapping solvers. We are currently working on optimally mapping the problem onto the restricted topology of the quantum annealer to improve the quality of the solution.